Download Sets Math 130 Linear Algebra

get away with “and so forth.” If you want to see
all of what “and so forth” entails, you can read
Dedekind’s 1888 paper Was sind und was sollen
die Zahlen? and my comments on it. In that article he starts off developing set theory and ends up
with the natural numbers.
The real numbers. These include all positive
numbers, negative numbers, and 0. Besides the natural numbers, their negations and 0 are included,
√
,
algebraic
numbers
like
5, and
fractions like 22
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transcendental numbers like π, e. If a number can
be named decimally with infinitely many digits,
then it’s a real number. We’ll use R to denote the
set of all real numbers. Like N, R has lots of operations and functions associated with it, but treated
as a set, all it has is its elements, the real numbers.
Note that N is a subset of R since every natural
number is a real number.
Sets
Math 130 Linear Algebra
D Joyce, Fall 2013
Just a little bit about sets. We’ll use the language of sets throughout the course, but we’re not
using much of set theory. Still, it would be useful
to know a little bit about it.
A set itself is just supposed to be something that
has elements. It doesn’t have to have any structure but just have elements. The elements can be
anything, but usually they’ll be things of the same
kind.
If you’ve only got one set, however, there’s no
need to even mention sets. It’s when several sets
are under consideration that the language of sets
becomes useful.
There are ways to construct new sets, too, and
these constructions are important. The most important of these is a way to collect some of the
elements in a set to form another set, a subset of
the first.
Subsets. If you have a set and a language to talk
about elements in that set, then you can form subsets of that set by properties of elements in that
language.
For instance, we have arithmetic on R, so solutions to equations are subsets of R. The solutions
to the equation x3 = x are 0, 1, and −1. We can
Examples. Let’s start with sets of numbers. describe its solution set using the notation
There are ways of constructing these sets, but let’s
S = {x ∈ R | x3 = x}
not deal with that now. Let’s assume that we already have these sets.
which is read as “S is the set of x in R such that
The natural numbers. These are the counting x3 = x.” We could also describe that set by listing
numbers, that is, whole positive numbers. 1 is the its elements, S = {0, 1, −1}. When you name a
first natural number, 2 the second, 3, the third, etc. set by listing its elements, the order that you name
We’ll use N to denote the set of all natural num- them doesn’t matter. We could have also written
bers. Some people like to include 0 in the natural S = {−1, 0, 1} for the same set.
Open and closed intervals in R are also subsets
numbers, but I follow Dedekind who started with
of
R. For example,
1. There is a structure on N, namely there are operations of addition, subtraction, etc., but as a set,
(3, 5) = {x ∈ R | 3 < x < 5}
it’s just the numbers. You’ll often see N defined as
[3, 5] = {x ∈ R | 3 ≤ x ≤ 5}
N = {1, 2, 3, . . .}
which is read as N is the set whose elements are 1, 2,
There are a couple of notations for subsets. We’ll
3, and so forth. That’s just an informal way of describing what N is. A complete description couldn’t use the notation A ⊆ S to say that A is a subset
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of S. We allow S ⊆ S, that is, we consider a set
DeMorgan’s laws describe a duality between inS to be a subset of itself. If a subset A doesn’t tersection and union. They can be written as
include all the elements of S, then A is called a
A∩B = A∪B
proper subset of S. The only subset of S that’s not
a proper subset is S itself. We’ll use the notation
A∪B = A∩B
A ⊂ S to indicate that A is a proper subset of S.
The distributivity laws say that intersection and
(Warning. There’s an alternate notational conunion
each distribute over the other
vention for subsets. In that notation A ⊂ S means
A is any subset of S, while A ( S means A is a
(A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C)
proper subset of S. I prefer the the notation we’re
(A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C)
using because it’s analogous to the notations ≤ for
less than or equal, and < for less than.)
Products of sets. So far we’ve looked at creatOperations on subsets. Frequently you deal ing sets within set. There are some operations on
with several subsets of a set, and there are oper- sets that create bigger sets, the most important beations of intersection, union, and difference that ing creating products of sets. These depend on the
describe new subsets in terms of previously known concept of ordered pairs of elements. The notation
for ordered pair (a, b) of two elements extends the
subsets.
The intersection A ∩ B of two subsets A and B usual notation we use for coordinates in the xyof a given set S is the subset of S that includes all plane. The important property of ordered pairs is
that two ordered pairs are equal if and only if they
the elements that are in both A and B:
have the same first and second coordinates:
A ∩ B = {x ∈ S | x ∈ A and x ∈ B}.
(a, b) = (c, d) iff a = c and b = d.
The union A ∪ B of two subsets A and B of a
The product of two sets S and T consists of all
given set S is the subset of S that includes all the
the
ordered pairs where the first element comes
elements that are in A or in B or in both:
from S and the second element comes from T :
A ∪ B = {x ∈ S | x ∈ A or x ∈ B}.
S × T = {(a, b) | a ∈ S and b ∈ T }.
As usual in mathematics, the word “or” means an
Thus, the usual xy-plane is R × R, usually deinclusive or and implicitly includes “or both.”
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The difference A − B of two subsets A and B of noted R .
Besides binary products S × T , you can analoa given set S is the subset of S that includes all the
gously define ternary products S × T × U in terms
elements that are in A but not in B:
of triples (a, b, c) where a ∈ S, b ∈ T , and c ∈ U ,
and higher products, too.
A − B = {x ∈ S | x ∈ A and x ∈
/ B}
There’s also the complement of a subset A of
a set S. The complement is just S − A, all the
elements of S that aren’t in A. When the set S is
understood, the complement of A often is denoted
more simply as either A or Ac .
These operations satisfy lots of identities. I’ll just
name a couple of important ones.
Sets of subsets; power sets. Another way to
create bigger sets is to form sets of subsets. If you
collect all the subsets of a given set S into a set,
then the set of all those subsets is called the power
set of S, denotes P(S) or sometimes 2S .
For example, let S be a set with 3 elements, S =
{a, b, c}. Then S has eight subsets. There are three
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singleton subsets, that is, subsets having exactly
one element, namely {a}, {b}, and {c}. There are
three subsets having exactly two elements, namely
{a, b}, {a, c}, and {b, c}. There’s one subset having
all three elements, namely S itself. And there’s one
subset that has no elements. You could denote it
{}, but it’s always denoted ∅ and called the empty
set or null set. Thus, the power set of S has eight
elements
P(S) = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, S}.
Functions and function sets. A function f :
S → T from a set S to a set T can be identified
with its graph. Its graph is a particular subset of
the product set S × T , namely, the subset
{(x, y) | x ∈ S and y = f (x)}.
You can tell which subsets A of S × T are graphs
of functions. They’re the ones with the following
property: for each x ∈ S, there is exactly ordered
pair in A whose first element is x.
It’s convenient to identify functions with such
graphs.
All the functions from S to T can be collected
together to form a set, sometimes called a function
set, and denoted either T S or F(S, T ). Since each
function is a subset of S × T , this function set is
actually a subset of the power set of S × T , that is,
F(S, T ) ⊆ P(S × T ).
Math 130 Home Page at
http://math.clarku.edu/~djoyce/ma130/
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